(a) p1x1* + p2x2* = y
U (x1,x2)x1 axb 2
is
( ) (x * 1 ,x * 2 )(a a m b )p 1 ,(a b m b )p 2.
Computing Ordinary Demands -
a Cobb-Douglas Example.
x2
U (x1,x2)x1 axb 2xFra bibliotek* 2bm
(a b )p 2
Chapter Five
Choice 消费者最优选择
Where Are We Doing in This Chapter?
After modeling a consumer’s choice set and his preference (represented by utility functions), we now put them together and model how he/she makes optimal choice.
So the MRS is
M R S d d x x 1 2 U U / /x x 1 2 b a x x 1 1 a a x 1 b 2 x b 2 1 a b x x 1 2 .
Computing Ordinary Demands a Cobb-Douglas Example.
In mathematical terms, this is a constrained maximization problem;
In economics, this is a rational choice problem.
Rational Constrained Choice
x2
More preferred bundles
(x1*,x2*) also exhausts the budget so
p1x* 1p2x* 2m . (B)
Computing Ordinary Demands a Cobb-Douglas Example.
So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences
Suppose that the consumer has Cobb-Douglas preferences.
Then
U (x1,x2)x1 axb 2
M U1x U 1ax1 a1xb 2
M U2x U 2bx1 axb 21
Computing Ordinary Demands a Cobb-Douglas Example.
curve at (x1*,x2*) equals the slope of the budget
constraint. x2*
x1*
x1
Rational Constrained Choice
(x1*,x2*) satisfies two conditions: (a) the budget is exhausted;
x2
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget; p1x1* + p2x2* = m.
x2*
x1*
x1
Rational Constrained Choice
x2
(x1*,x2*) is interior .
(b) The slope of the indiff.
Computing Ordinary Demands a Cobb-Douglas Example.
Suppose that the consumer has Cobb-Douglas preferences.
U (x1,x2)x1 axb 2
Computing Ordinary Demands a Cobb-Douglas Example.
So the MRS is
M R S d d x x 1 2 U U / /x x 1 2 b a x x 1 1 a a x 1 b 2 x b 2 1 a b x x 1 2 .
At
(x1*,x2*),
a b
MRpS1=x-1p*1/p2
p2
x
* 2
so
(A)
Computing Ordinary Demands a Cobb-Douglas Example.
Rational Constrained Choice
When x1* > 0 and x2* > 0 the demanded bundle is INTERIOR. If buying (x1*,x2*) costs $m then the budget is exhausted.
Rational Constrained Choice
x*1
am (ab)p1
x1
Rational Constrained Choice
When x1* > 0 and x2* > 0 and (x1*,x2*) exhausts the budget, and indifference curves have no
‘kinks’, the ordinary demands are obtained by solving:
p1x1* + p2x2* = m (b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).
Affordable bundles
x1
Rational Constrained Choice
The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and budget. Ordinary demands will be denoted by x1*(p1,p2,m) and x2*(p1,p2,m).